All Topics

Acute
square triangulation. Can one partition the square into triangles with all
angles acute? How many triangles are needed, and what is the best angle bound
possible?

Adventitious
geometry. Quadrilaterals in which the sides and diagonals form more
rational angles with each other than one might expect. Dave Rusin's known math
pages include another
article on the same problem.

1st
and 2nd Ajima-Malfatti points. How to pack three circles in a triangle so
they each touch the other two and two triangle sides. This problem has a
curious history, described in Wells' Penguin Dictionary of Curious and
Interesting Geometry: Malfatti's original (1803) question was to carve
three columns out of a prism-shaped block of marble with as little wasted
stone as possible, but it wasn't until 1967 that it was shown that these three
mutually tangent circles are never the right answer. See also this
Cabri geometry page.

On angles
whose squared trigonometric functions are rational, J. Conway, C. Radin,
and L. Sadun. This somewhat technical paper on the theory of Dehn invariants
(used to determine whether there exists a dissection from one polyhedron to
another) makes the theory more computationally effective. It contains the
fascinating observation that there should exist a dissection that combines
pieces from a dodecahedron, icosahedron, and icosidodecahedron to form a
single large cube. How many pieces are needed?

Ant on a block. If
you walk along the surface of a 1x1x2 rectangular block, from one corner,
where is the farthest point? You would think the opposite corner, right?

Antipodes.
Jim Propp asks whether the two farthest apart points, as measured by surface
distance, on a symmetric convex body must be opposite each other on the body.
Apparently this is open even for rectangular boxes.

Area of
the Mandelbrot set. One can upper bound this area by filling the area
around the set by disks, or lower bound it by counting pixels; strangely, Stan
Isaacs notes, these two methods do not seem to give the same answer.

Arranging six
squares. This Geometry Forum problem of the week asks for the number of
different hexominoes, and for how many of them can be folded into a cube.

ARTiP: an
automated rectangular tiling prover. This system uses a constraint-propagation
algorithm, similar to Waltz' famous line-labeling technique, to automatically
find dissections of planar regions into rectangles.

The bellows
conjecture, R. Connelly, I. Sabitov and A. Walz in Contributions to
Algebra and Geometry , volume 38 (1997), No.1, 1-10. Connelly had previously
discovered non-convex polyhedra which are flexible (can move through a
continuous family of shapes without bending or otherwise deforming any faces);
these authors prove that in any such example, the volume remains constant
throughout the flexing motion.

Belousov's Brew. A
recipe for making spiraling patterns in chemical reactions.

Blocking
polyominos. R. M. Kurchan asks, for each k, what is the smallest polyomino
such that k copies can form a "blocked" configuration in which no piece can be
slid free of the others, but in which any subconfiguration is not blocked.

Bounded
degree triangulation. Pankaj Agarwal and Sandeep Sen ask for
triangulations of convex polytopes in which the vertex or edge degree is
bounded by a constant or polylog.

Box in
a box. What is the smallest cube that can be put inside another cube
touching all its faces? There is a simple solution, but it seems difficult to
prove its correctness. The solution and proof are even prettier in four
dimensions.

A Brunnian
link. Cutting any one of five links allows the remaining four to be
disconnected from each other, so this is in some sense a generalization of the
Borromean rings. However since each pair of links crosses four times, it can't
be drawn with circles.

Buckyballs.
The truncated icosahedron recently acquired new fame and a new name when
chemists discovered that Carbon forms molecules with its shape.

Building a
better beam detector. This is a set that intersects all lines through the
unit disk. The construction below achieves total length approximately 5.1547,
but better bounds were previously known.

Centers
of maximum matchings. Andy Fingerhut asks, given a maximum (not minimum)
matching of six points in the Euclidean plane, whether there is a center point
close to all matched edges (within distance a constant times the length of the
edge). If so, it could be extended to more points via Helly's theorem.
Apparently this is related to communication network design. I include a
response I sent with a proof (of a constant worse than the one he wanted, but
generalizing as well to bipartite matching).

The
Cheng-Pleijel point. Given a closed plane curve and a height H, this point
is the apex of the minimum surface area cone of height H over the curve. Ben
Cheng demonstrates this concept with the help of a Java applet.

The
chromatic number of the plane. Gordon Royle and Ilan Vardi summarize
what's known about the famous open problem of how many colors are needed to
color the plane so that no two points at a unit distance apart get the same
color. See also another
article from Dave Rusin's known math pages.

Circle fractal based on
repeated placement of two equal tangent circles within each circle of the
figure. One could also get something like this by inversion, starting with
three mutually tangent circles, but then the circles at each level of the
recursion wouldn't all stay the same size as each other.

Circular
coverage constants. How big must N equal disks be in order to completely
cover the unit disk? What about disks with sizes in geometric progression?
From MathSoft's favorite constants pages.

Circular
quadrilaterals. Bill Taylor notes that if one connects the opposite
midpoints of a partition of the circle into four chords, the two line segments
you get are at right angles. Geoff Bailey supplies an elegant proof.

A
computational approach to tilings. Daniel Huson investigates the
combinatorics of periodic tilings in two and three dimensions, including a
classification of the tilings by shapes topologically equivalent to the five
Platonic solids.

Covering
points by rectangles. Stan Shebs discusses the problem of finding a
minimum number of copies of a given rectangle that will cover all points in
some set, and mentions an application to a computer strategy game. This is
NP-hard, but I don't know how easy it is to approximate; most related work I
know of is on optimizing the rectangle size for a cover by a fixed number of
rectangles.

Crystallographic
topology. C. Johnson and M. Burnett of Oak Ridge National Lab use
topological methods to understand and classify the symmetries of the lattice
structures formed by crystals. (Somewhat technical.)

CSE
logo. This java applet allows interactive control of a rotating collection
of cubes.

Cube
Dissection. How many smaller cubes can one divide a cube into? From Eric
Weisstein's treasure trove of mathematics.

Curvature of
crossing convex curves. Oded Schramm considers two smooth convex planar
curves crossing at at least three points, and claims that the minimum
curvature of one is at most the maximum curvature of the other. Apparently
this is related to conformal mapping. He asks for prior appearances of this
problem in the literature.

Curvature
of knots. Steve Fenner shows that any smooth, simple, closed curve in
3-space must have total curvature at least 4 pi.

Cut-the-knot logo.
With a proof of the origami-folklore that this folded-flat overhand knot forms
a regular pentagon.

Dehn invariants
of hyperbolic tiles. The Dehn invariant is one way of testing whether a
Euclidean polyhedron can be used to tile space. But as Doug Zare describes,
there are hyperbolic tiles with nonzero Dehn invariant.

Delaunay
and regular triangulations. Lecture by Herbert Edelsbrunner, transcribed
by Pedro Ramos and Saugata Basu. The regular triangulation has been
popularized by Herbert as the appropriate generalization of the Delaunay
triangulation to collections of disks.

Digital
Diffraction, B. Hayes, Amer. Scientist 84(3), May-June 1996. What
does the Fourier transform of a geometric figure such as a regular pentagon
look like? The answer can reveal symmetries of interest to crystallographers.

Dilation-free
planar graphs. How can you arrange n points so that the set of all lines
between them forms a planar graph with no extra vertices?

Direct
opposite Reverse. David Sterner claims to have invented one of "the six
simplest known solids to be mathematically defined" and uses its chromatic
aberration in a 3d-photograph process for the seven-eyed.

Disjoint
triangles. Any 3n points in the plane can be partitioned into n disjoint
triangles. A. Bogomolny gives a simple proof and discusses some
generalizations.

An
equilateral dillemma. IBM asks you to prove that the only triangles that
can be circumscribed around an equilateral triangle, with their vertices
equidistant from the equilateral vertices, are themselves equilateral.

Equilateral
triangles. Dan Asimov asks how large a triangle will fit into a square
torus; equivalently, the densest packing of equilateral triangles in the
pattern of a square lattice. There is only one parameter to optimize, the
angle of the triangle to the lattice vectors; my answer is
that the densest packing occurs when this angle is 15 or 45 degrees, shown below.
(If the lattice doesn't have to be square, it is possible to get density 2/3;
apparently this was long known, e.g. see Fáry, Bull. Soc. Math. France 78
(1950) 152.)

Asimov also asks for the smallest triangle that will always cover at least
one point of the integer lattice, or equivalently a triangle such that no
matter at what angle you place copies of it on an integer lattice, they always
cover the plane; my guess is that the worst angle is parallel and 30 degrees
to the lattice, giving a triangle with 2-unit sides and contradicting an
earlier answer to Asimov's question.

Filling
space with unit circles. Daniel Asimov asks what fraction of 3-dimensional
space can be filled by a collection of disjoint unit circles. (It may not be
obvious that this fraction is nonzero, but a standard construction allows one
to construct a solid torus out of circles, and one can then pack tori to fill
space, leaving some uncovered gaps between the tori.) The geometry center has
information in several places on this problem, the best being an article
describing a way of filling space by unit circles (discontinuously).

Five
circle theorem. Karl Rubin and Noam Elkies asked for a proof that a
certain construction leads to five cocircular points. This result was
subsequently discovered by Allan Adler and Gerald Edgar to be essentially the
same as a theorem proven in 1939 by F. Bath.

The Fourth
Dimension. John Savard provides a nice graphical explanation of the
four-dimensional regular polytopes.

Four-dimensional
visualization. Doug Zare gives some pointers on high-dimensional
visualization including a description of an interesting chain of successively
higher dimensional polytopes beginning with a triangular prism.

A fractal
beta-skeleton with high dilation. Beta-skeletons are graphs used, among
other applications, in predicting which pairs of cities should be connected by
roads in a road network. But if you build your road network this way, it may
take you a long time to get from point a to point b.

Fractals. The spanky fractal
database at Canada's national meson research facility.

Fractiles, multicolored magnetic
rhombs with angles based on multiples of pi/7.

Fractional Graph
Theory, a rational approach to the theory of graphs, Edward R. Scheinerman
and Daniel Ullman, Johns Hopkins. Explains why the fractional chromatic number
of the plane is at most 7 and at least 32/9.

Erich
Friedman's dissection puzzle. Partition a 21x42x42 isosceles triangles
into six smaller triangles, all similar to the original but with no two equal
sizes. (The link is to a drawing of the solution.)

Gallery of interactive
on-line geometry. The Geometry Center's collection includes programs for
generating Penrose tilings, making periodic drawings a la Escher in the
Euclidean and hyperbolic planes, playing pinball in negatively curved spaces,
viewing 3d objects, exploring the space of angle geometries, and visualizing
Riemann surfaces.

Gauss'
tomb. The story that he asked for (and failed to get) a regular 17-gon
carved on it leads to some discussion of 17-gon construction and perfectly
scalene triangles.

Graham's
hexagon, maximizing the ratio of area to diameter. You'd expect it to be a
regular hexagon, right? Wrong. From MathSoft's favorite constants pages. See
also Wolfgang
Schildbach's java animation of this hexagon and similar n-gons for larger
values of n.

The
golden ratio in an equilateral triangle. If one inscribes a circle in an
ideal hyperbolic triangle, its points of tangency form an equilateral triangle
with side length 4 ln phi! One can then place horocycles centered on
the ideal triangle's vertices and tangent to each side of the inner
equilateral triangle. From the Cabri geometry site. (In French.)

The golden
spiral. This shape, constructed by inscribing circular arcs in a spiral tiling
of squares, resembles but is not quite the same as a logarithmic spiral. A
similar spiral is used as the Sybase Inc.
logo.

Golden
spiral jewelry image made by Keith Halewood to commemorate his Welsh
heritage. This one combines three Archimedean spirals and doesn't have
anything to do with the golden ratio. The meaning of the triple spiral symbol
is explained by this
note.

Ham
Sandwich Theorem: you can always cut your ham and two slices of bread each
in half with one slice, even before putting them together into a sandwich.
From Eric Weisstein's treasure trove of mathematics.

Heesch's
problem. How many times can a shape be completely surrounded by copies of
itself, without being able to tile the entire plane? W. R. Marshall and C.
Mann have recently made significant progress on this problem using shapes
formed by indenting and outdenting the edges of polyhexes.

Heureka,
the Finnish science center uses Penrose tiles to pave the area in front of
its main entrance. (Unfortunately, the picture included here is not very good
-- see the Mathematical Intelligencer 18(4), Fall 1996, p. 65 for a
better photo.)

Hexagon
tiling. The regular tiling by hexagons can be repeatedly subdivided and
recombined into a tiling by hexagons 1/7 the size of the original, to form an
interesting recursive structure. From Paul Bourke's
geometry page.

How many
points can one find in three-dimensional space so that all triangles are
equilateral or isosceles? One eight-point solution is formed by placing three
points on the axis of a regular pentagon. This problem seems related to the
fact that any
planar point set forms O(n7/3) isosceles triangles; in three
dimensions, Theta(n3) are possible (by generalizing the pentagon
solution above). From Stan Wagon's PotW archive.

Hyperbolic
packing of convex bodies. William Thurston answers a question of Greg Kuperberg, on whether there
is a constant C such that every convex body in the hyperbolic plane can be
packed with density C. The answer is no -- long skinny bodies can not be
packed efficiently.

The
hyperbolic surface activity page. Tom Holroyd describes hyperbolic
surfaces occurring in nature, and explains how to make a paper model of a
hyperbolic surface based on a tiling by heptagons.

Integer
distances. Robert Israel gives a nice proof (originally due to Erdös) of
the fact that, in any non-colinear planar point set in which all distances are
integers, there are only finitely many points. Infinite sets of points with
rational distances are known, from which arbitrarily large finite sets of
points with integer distances can be constructed; however it is open whether
there are even seven points at integer distances in general position (no three
in a line and no four on a circle).

Intersecting
cube diagonals. Mark McConnell asks for a proof that, if a convex
polyhedron combinatorially equivalent to a cube has three of the four body
diagonals meeting at a point, then the fourth one meets there as well. There
is apparently some connection to toric varieties.

Isoperimetric
polygons. Livio Zucca groups grid polygons by their perimeter instead of
by their area. For small integer perimeter the results are just polyominos but
after that it gets more complicated...

Isosceles
pairs. Stan Wagon asks which triangles can be dissected into two isosceles
triangles.

Jordan
sorting. This is the problem of sorting (by x-coordinate) the
intersections of a line with a simple polygon. Complicated linear time
algorithms for this are known (for instance one can triangulate the polygon
then walk from triangle to triangle); Paul Callahan discusses an alternate
algorithm based on the dynamic optimality conjecture for splay trees.

Kelvin
conjecture counterexample. Evelyn Sander forwards news about the discovery
by Phelan and Weaire of a better way to partition space into equal-volume
low-surface-area cells. Kelvin had conjectured that the truncated octahedron
provided the optimal solution, but this turned out not to be true.

Looking at
sunflowers. In this abstract of an undergraduate research paper, Surat
Intasang investigates the spiral patterns formed by sunflower seeds, and
discovers that often four sets of spirals can be discerned, rather than the
two sets one normally notices.

The
Margulis Napkin Problem. Jim Propp asked for a proof that the perimeter of
a flat origami figure must be at most that of the original starting square.
Gregory Sorkin provides a simple example showing that on the contrary, the
perimeter can be arbitrarily large.

Martin's pretty
polyhedra. Simulation of particles repelling each other on the sphere
produces nice triangulations of its surface.

Match sticks in
the summer. Ivars Peterson discusses the graphs that can be formed by
connecting vertices by non-crossing equal-length line segments.

Mirror
Curves. Slavik Jablan investigates patterns formed by crisscrossing a
curve around points in a regular grid, and finds examples of these patterns in
art from various cultures.

Mirrored
room illumination. A summary by Christine Piatko of the old open problem
of, given a polygon in which all sides are perfect mirrors, and a point source
of light, whether the entire polygon will be lit up. The answer is no if
smooth curves are allowed. See also Eric Weisstein's page on the Illumination
Problem.

Mitre Tiling. Ed Pegg
describes the discovery of the versatile tiling system (with Adrian Fisher and
Miroslav Vicher), also discussing many other interesting tilings including a
tile that can fill the plane with either five-fold or six-fold symmetry.

Models of
Small Geometries. Burkard Polster draws diagrams of combinatorial
configurations such as the Fano plane and Desargues' theorem (shown below) in
an attempt to capture the mathematical beauty of these geometries.

Monge's
theorem and Desargues' theorem, identified. Thomas Banchoff relates these
two results, on colinearity of intersections of external tangents to disjoint
circles, and of intersections of sides of perspective triangles, respectively.
He also describes generalizations to higher dimensional spheres.

No
cubed cube. David Moews offers a cute proof that no cube can be divided
into smaller cubes, all different.

The
no-three-in-line problem. How many points can be placed in an
n*n grid with no three on a common line? The solution is known
to be between 1.5n and 2n. Achim Flammenkamp discusses some new
computational results including bounds on the number of symmetric solutions.

Non-Euclidean
geometry with LOGO. A project at Cardiff, Wales, for using the LOGO
programming language to help mathematics students visualise non-Euclidean
geometry.

Not.
AMS Cover, Apr. 1995. This illustration for an article on geometric
tomography depicts objects (a cuboctahedron and warped rhombic dodecahedron)
that disguise themselves as regular tetrahedra by having the same width
function or x-ray image.

Odd rectangles for
L4n+2. Phillippe Rosselet shows that any L-shaped (4n+2)-omino
can tile a rectangle with an odd side.

Odd
squared distances. Warren Smith considers point sets for which the square
of each interpoint distance is an odd integer. Clearly one can always do this
with an appropriately scaled regular simplex; Warren shows that one can
squeeze just one more point in, iff the dimension is 2 (mod 4). Moshe
Rosenfeld has published a related paper in Geombinatorics (vol. 5, 1996, pp.
156-159).

Origamic
tetrahedron. The image below depicts a way of making five folds in a 2-3-4
triangle, so that it folds up into a tetrahedron. Toshi Kato asks if you can
fold the triangle into a tetrahedron with only three folds. It turns out that
there is a unique solution, although many tetrahedra can be formed with more
folds.

Parallel
pentagons. Thomas Feng defines these as pentagons in which each diagonal
is parallel to its opposite side, and asks for a clean construction of a
parallel pentagon through three given points. (He is aware of the obvious
reduction via affine transformation to the construction of regular pentagons,
but finds that non-elegant.)

Penrose
tiles and worse. This article from Dave Rusin's known math pages discusses
the difficulty of correctly placing tiles in a Penrose tiling, as well as
describing other tilings such as the pinwheel.

Penrose-Wang
tilings. Tony Smith describes some of the mathematics behind these
aperiodic tilings, somehow leading to the concluding question "Can musical
sequences also simulate the operation of any Turing machine?"

Person
polygons. Marc van Kreveld defines this interesting and important class of
simple polygons, and derives a linear time algorithm (with a rather large
constant factor) for recognizing a special case in which there are many reflex
vertices.

Polygons
as projections of polytopes. Andrew Kepert answers a question of George
Baloglou on whether every planar figure formed by a convex polygon and all its
diagonals can be formed by projecting a three-dimensional convex polyhedron.

Polyhedra.
Bruce Fast is building a library of images of polyhedra. He describes some of
the regular and semi-regular polyhedra, and lists names of many more including
the Johnson solids (all convex polyhedra with regular faces).

Polyiamonds.
This Geometry Forum problem of the week asks whether a six-point star can be
dissected to form eight distinct hexiamonds.

PolyMultiForms.
L. Zucca uses pinwheel tilers to dissect an illustration of the Pythagorean
theorem into few congruent triangles.

Polyomino covers.
Alexandre Owen Muniz investigates the minimum size of a polygon that can
contain each of the n-ominoes.

Polyomino
inclusion problem. Yann David wants to know how to test whether all
sufficiently large polyominoes contain at least one member of a given set.

Polyomino problems
and variations of a theme. Information about filling rectangles, other
polygons, boxes, etc., with dominoes, trominoes, tetrominoes, pentominoes,
solid pentominoes, hexiamonds, and whatever else people have invented as
variations of a theme.

Polyomino
tiling. Joseph Myers classifies the n-ominoes up to n=15 according to how
symmetrically they can tile the plane.

Polyominoes,
figures formed from subsets of the square lattice tiling of the plane.
Interesting problems associated with these shapes include finding all of them,
determining which ones tile the plane, and dissecting rectangles or other
shapes into sets of them. Also includes related material on polyiamonds,
polyhexes, and animals.

Postscript
geometry. Bill Casselman uses postscript to motivate a course in Euclidean
geometry. See also his Coxeter group graph
paper. and Phil Smith's Postscript
Doodles page, especially the postscript spirograph. Beware, however, that
postscript can not really represent such basic geometric primitives as
circles, instead approximating them by splines.

Quaquaversal
Tilings and Rotations. John Conway and Charles Radin describe a
three-dimensional generalization of the pinwheel tiling, the mathematics of
which is messier due to the noncommutativity of three-dimensional rotations.

Quasicrystals
and color symmetry. Ron Lifshitz provides a light introduction to the
symmetry of periodic and aperiodic crystals, and the complications introduced
by including permutations of colors in a coloring as part of a symmetry
operation. His publication list
includes more technical material on the same subject.

A
quasi-polynomial bound for the diameter of graphs of polyhedra, G. Kalai
and D. Kleitman, Bull. AMS 26 (1992). A famous open conjecture in polyhedral
combinatorics (with applications to e.g. the simplex method in linear
programming) states that any two vertices of an n-face polytope are linked by
a chain of O(n) edges. This paper gives the weaker bound O(nlog d).

Rational
triangles. This well known problem asks whether there exists a triangle
with the side lengths, medians, altitudes, and area all rational numbers.
Randall Rathbun provides some "near misses" -- triangles in which most but not
all of these quantities are irrational. See also Dan Asimov's question
in geometry.puzzles about integer right-angled tetrahedra.

Realizing a
Delaunay triangulation. Many authors have written Java code for computing
Delaunay triangulations of points. But Tim Lambert's applet does the reverse:
give it a triangulation, and it finds points for which that triangulation is
Delaunay.

Rhombic
tilings. Abstract of Serge Elnitsky's thesis, "Rhombic tilings of polygons
and classes of reduced words in Coxeter groups". He also supplied the picture
below of a rhombically tiled 48-gon, available with better color resolution
from his website.

Rigid
regular r-gons. Erich Friedman asks how many unit-length bars are needed
in a bar-and-joint linkage network to make a unit regular polygon rigid. What
if the polygon can have non-unit-length edges?

Rubik's hypercube.
3x3x3x3 times as much puzzlement. Windows software from Daniel Green and Don
Hatch, now also available as Linux executable and C++ source.

Rudin's example of
an unshellable triangulation. In this subdivision of a big tetrahedron
into small tetrahedra, every small tetrahedron has a vertex interior to a face
of the big tetrahedron, so you can't remove any of them without forming a
hole. Peter Alfeld, Utah.

Sacred
Geometry. Mystic insights into the "principle of oneness underlying all
geometry", mixed with occasional outright falsehoods such as the suggestion
that dodecahedra and icosahedra arise in crystals. But the illustrative
diagrams are ok, if you just ignore the words... For more mystic diagrams, see
The Sacred Geometry
Coloring Book.

Saints Among
Us. Anna Chupa makes kaleidoscopic photomontages based on the geometry of
the Penrose tiling.

Santa Rosa
Menger Cube made by Tom Falbo and helpers at Santa Rosa Junior College
from 8000 1"-cubed oak blocks.

Satellite
constellations. Sort of a dynamic version of a sphere packing problem: how
to arrange a bunch of satellites so each point of the planet can always see
one of them?

Sausage
Conjecture. L. Fejes Tóth conjectured that, to minimize the volume of the
convex hull of hyperspheres in five or more dimensions, one should line them
up in a row. This has recently been solved for very high dimensions (d
> 42) by Betke
and Henk (see also Betke et al., J. Reine Angew. Math. 453 (1994)
165-191).

The
Schläfli Double Six. A lovely photo-essay of models of this configuration,
in which twelve lines each meet five of thirty points. (This site also refers
to related configurations involving 27 lines meeting either 45 or 135 points,
but doesn't describe any mathematical details. For further descriptions of all
of these, see Hilbert and Cohn-Vossen's "Geometry and the Imagination".)

Sighting
point. John McKay asks, given a set of co-planar points, how to find a
point to view them all from in a way that maximizes the minimum viewing angle
between any two points. Somehow this is related to monodromy groups. I don't
know whether he ever got a useful response. This is clearly polynomial time:
the decision problem can be solved by finding the intersection of
O(n2) shapes, each the union of two disks, so doing this naively
and applying parametric search gives O(n4 polylog), but it might be
interesting to push the time bound further. A closely related problem of smoothing
a triangular mesh by moving points one at a time to optimize the angles of
incident triangles can be solved in linear time by LP-type algorithms
[Matousek, Sharir, and Welzl, SCG 1992; Amenta, Bern,
and Eppstein, SODA 1997].

Simplex/hyperplane
intersection. Doug Zare nicely summarizes the shapes that can arise on
intersecting a simplex with a hyperplane: if there are p points on the
hyperplane, m on one side, and n on the other side, the shape is (a projective
transformation of) a p-iterated cone over the product of m-1 and n-1
dimensional simplices.

Six-regular
toroid. Mike Paterson asks whether it is possible to make a torus-shaped
polyhedron in which exactly six equilateral triangles meet at each vertex.

Skewered
lines. Jim Buddenhagen notes that four lines in general position in
R3 have exactly two lines crossing them all, and asks how this
generalizes to higher dimensions.

Sketchpad
demo includes a Reuleaux triangle rolling between two parallel lines.

Sliceforms,
3d models made by interleaving two directions of planar slices.

Sphere
packing and kissing numbers. How should one arrange circles or spheres so
that they fill space as densely as possible? What is the maximum number of
spheres that can simultanously touch another sphere?

Spherical Julia set with dodecahedral symmetry discovered by McMullen and
Doyle in their work on quintic
equations and rendered by Don Mitchell. Update 12/14/00: I've lost the big
version of this image and can't find DonM anywhere on the net -- can anyone
help? In the meantime, here's a link to McMullen's
rendering.

Squares on
a Jordan curve. Various people discuss the open problem of whether any
Jordan curve in the plane contains four points forming the vertices of a
square, and the related but not open problem of how to place a square table
level on a hilltop. This is also in the geometry.puzzles
archive.

Structors. Panagiotis
Karagiorgis thinks he can get people to pay large sums of money for exclusive
rights to use four-dimensional regular polytopes as building floor plans. But
he does have some pretty pictures...

Sums
of square roots. A major bottleneck in proving NP-completeness for
geometric problems is a mismatch between the real-number and Turing machine
models of computation: one is good for geometric algorithms but bad for
reductions, and the other vice versa. Specifically, it is not known on Turing
machines how to quickly compare a sum of distances (square roots of integers)
with an integer or other similar sums, so even (decision versions of) easy
problems such as the minimum spanning tree are not known to be in NP. Joe O'Rourke discusses an approach to
this problem based on bounding the smallest difference between two such sums,
so that one could know how precise an approximation to compute.

Sylvester's
theorem. This states that any finite non-colinear point set has a line
containing only two points (equivalently, every zonohedron has a quadrilateral
face). Michael Larsen, Tim Chow, and Noam Elkies discuss two proofs and a
complex-number generalization. (They omit the very simple generalization from
Euler's
formula: every convex polyhedron has a face of degree at most five.)

These two pictures were orphaned when maths with photographs went
offline. Does anyone know what places they are pictures of? (For another view
of the cuboctohedron sculpture, see Rod's cuboctahedron page.)

Thrackles
are graphs embedded as a set of curves in the plane that cross each other
exactly once; Conway has conjectured that an n-vertex thrackle has at
most n edges. Stephan Wehner describes what is known about thrackles.

Three-color
the Penrose tiling? Mark Bickford asks if this tiling is always
three-colorable. Ivars Peterson
reports on a new proof by Tom Sibley and Stan Wagon that the rhomb version
of the tiling is, but it's not clear whether this applies to the kites and
darts version. This is closely related to my page on line arrangement
coloring, since every Penrose tiling is dual to a "multigrid", which is
just an arrangement of lines in parallel families. But my page only deals with
finite arrangements, while Penrose tilings are infinite.

Three
cubes to one. Calydon asks whether nine pieces is optimal for this
dissection problem.

The Thurston
Project: experimental differential geometry, uniformization and quantum
field theory. Steve Braham hopes to prove Thurston's uniformization conjecture
by computing flows that iron the wrinkles out of manifolds.

Tic tac
toe theorem. Bill Taylor describes a construction of a warped tic tac toe
board from a given convex quadrilateral, and asks for a proof that the middle
quadrilateral has area 1/9 the original. Apparently this is not even worth a
chocolate fish.

A tiling
from ell. Stan Wagon asks which rectangles can be tiled with an
ell-tromino.

Tiling the
infinite grid with finite clusters. Mario Szegedy describes an algorithm
for determining whether a (possibly disconnected) polyomino will tile the
plane by translation, in the case where the number of squares in the polyomino
is a prime or four.

Tiling
with four cubes. Torsten Sillke summarizes results and conjectures on the
problem of tiling 3-dimensional boxes with a tile formed by gluing three cubes
onto three adjacent faces of a fourth cube.

Tiling with
notched cubes. Robert Hochberg and Michael Reid exhibit an unboxable
reptile: a polycube that can tile a larger copy of itself, but can't tile any
rectangular block.

Tobi Toys sell the
Vector Flexor, a flexible cuboctahedron skeleton, and Fold-a-form, an origami
business card that folds to form a tetrahedron that can be used as the
building block for more complex polyhedra.

Triangulation
numbers. These classify the geometric structure of viruses. Many viruses
are shaped as simplicial polyhedra consisting of 12 symmetrically placed
degree five vertices and more degree six vertices; the number represents the
distance between degree five vertices.

Triangulations
and arrangements. Two lectures by Godfried Toussaint, transcribed by Laura
Anderson and Peter Yamamoto. I only have the lecture on triangulations.

Triangulations
with many different areas. Eddie Grove asks for a function t(n) such that
any n-vertex convex polygon has a triangulation with at least t(n) distinct
triangle areas, and also discusses a special case in which the vertices are
points in a lattice.

Two-distance
sets. Timothy Murphy and others discuss how many points one can have in an
n-dimensional set, so that there are only two distinct interpoint distances.
The correct answer turns out to be n2/2 + O(n). This talk
abstract by Petr Lisonek (and paper in JCTA 77 (1997) 318-338) describe
some related results.

Unfold
the polygon. Olivier Devillers asks, if one is given a simple polygon,
treated as a linkage of rigid rods connected by hinges, can it be opened out
into a convex polygon without crossing itself?

Unfolding
the tesseract. Peter Turney lists the 261 polycubes that can be folded in
four dimensions to form the surface of a hypercube.

Unfurling crinkly
shapes. Science News discusses a recent result of Demaine, Connelly, and
Rote, that any nonconvex planar polygon can be continuously unfolded into
convex position.

Uniqueness of
focal points. A focal point (aka equichord) in a star-shaped curve is a
point such that all chords through the point have the same length. Noam Elkies
asks whether it is possible to have more than one focal point, and Curtis
McMullen discusses a generalization to non-star-shaped curves. This problem
has recently been put to rest by Marek Rychlik.

Voronoi
diagrams of lattices. Greg
Kuperberg discusses an algorithm for constructing the Voronoi cells in a
planar lattice of points. This problem is closely related to some important
number theory: Euclid's algorithm for integer GCD's, continued fractions, and
good approximations of real numbers by rationals. Higher-dimensional
generalizations (in which the Voronoi cells form zonotopes) are much harder --
one can find a basis of short vectors using the well-known LLL algorithm, but
this doesn't necessarily find the vectors corresponding to Voronoi
adjacencies. (In fact, according to Schattschneider's Quasicrystals and
Geometry, although the set of Voronoi adjacencies of any lattice generates
the lattice, it's not known whether this set always contains a basis.)

When
can a polygon fold to a polytope? A. Lubiw and J. O'Rourke describe
algorithms for finding the folds that turn an unfolded paper model of a
polyhedron into the polyhedron itself. It turns out that the familiar cross
hexomino pattern for folding cubes can also be used to fold three other
polyhedra with four, five, and eight sides.

Why
doesn't Pick's theorem generalize? One can compute the volume of a
two-dimensional polygon with integer coordinates by counting the number of
integer points in it and on its boundary, but this doesn't work in higher
dimensions.

Why
"snub cube"? John Conway provides a lesson on polyhedron nomenclature and
etymology. From the geometry.research archives.

Wonders
of Ancient Greek Mathematics, T. Reluga. This term paper for a course on
Greek science includes sections on the three classical problems, the
Pythagorean theorem, the golden ratio, and the Archimedean spiral.

A word
problem. Group theoretic mathematics for determining whether a polygon
formed out of hexagons can be dissected into three-hexagon triangles, or
whether a polygon formed out of squares can be dissected into
restricted-orientation triominoes.

WWW spirograph.
Fill in a form to specify radii, and generate pictures by rolling one circle
around another. For more pictures of cycloids, nephroids, trochoids, and
related spirograph shapes, see David Joyce's Little
Gallery of Roulettes, and the postscript spirograph machine on Phil
Smith's Postscript
Doodles page. Anu Garg has
implemented spirographs in Java.

Zometool. The 31-zone structural
system for constructing "mathematical models, from tilings to hyperspace
projections, as well as molecular models of quasicrystals and fullerenes, and
architectural space frame structures".